Factor Group-Sparse Regularization for Efficient Low-Rank Matrix Recovery [article]

Jicong Fan, Lijun Ding, Yudong Chen, Madeleine Udell
2019 arXiv   pre-print
This paper develops a new class of nonconvex regularizers for low-rank matrix recovery. Many regularizers are motivated as convex relaxations of the matrix rank function. Our new factor group-sparse regularizers are motivated as a relaxation of the number of nonzero columns in a factorization of the matrix. These nonconvex regularizers are sharper than the nuclear norm; indeed, we show they are related to Schatten-p norms with arbitrarily small 0 < p ≤ 1. Moreover, these factor group-sparse
more » ... larizers can be written in a factored form that enables efficient and effective nonconvex optimization; notably, the method does not use singular value decomposition. We provide generalization error bounds for low-rank matrix completion which show improved upper bounds for Schatten-p norm reglarization as p decreases. Compared to the max norm and the factored formulation of the nuclear norm, factor group-sparse regularizers are more efficient, accurate, and robust to the initial guess of rank. Experiments show promising performance of factor group-sparse regularization for low-rank matrix completion and robust principal component analysis.
arXiv:1911.05774v2 fatcat:eojp5fxderaejjx3j55b72rtqq